Constraint camera to move with object's bone
A ChildOf constraint allows you to parent an object to another with the ability to turn off some of the transformations that are copied.

To have the camera move with the character and allow the head/shoulders to rotate within the camera view, you turn off the rotation options in the constraint. You just need to pick a bone that you want to follow, maybe the shoulders or hips depending on how much of the character you want to move in front of the camera.

When you first choose the bone, the camera may jump out of place, click the Set Inverse button to fix this. You can also move the camera into position after you add the child of constraint



Here the camera has a ChidOf constrained set to the selected bone, you can see it follows the bottom point of the bone but doesn't rotate with it.